Integrand size = 20, antiderivative size = 68 \[ \int \frac {\left (c x^2\right )^{3/2}}{x (a+b x)^2} \, dx=\frac {c \sqrt {c x^2}}{b^2}-\frac {a^2 c \sqrt {c x^2}}{b^3 x (a+b x)}-\frac {2 a c \sqrt {c x^2} \log (a+b x)}{b^3 x} \]
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Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 45} \[ \int \frac {\left (c x^2\right )^{3/2}}{x (a+b x)^2} \, dx=-\frac {a^2 c \sqrt {c x^2}}{b^3 x (a+b x)}-\frac {2 a c \sqrt {c x^2} \log (a+b x)}{b^3 x}+\frac {c \sqrt {c x^2}}{b^2} \]
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Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c \sqrt {c x^2}\right ) \int \frac {x^2}{(a+b x)^2} \, dx}{x} \\ & = \frac {\left (c \sqrt {c x^2}\right ) \int \left (\frac {1}{b^2}+\frac {a^2}{b^2 (a+b x)^2}-\frac {2 a}{b^2 (a+b x)}\right ) \, dx}{x} \\ & = \frac {c \sqrt {c x^2}}{b^2}-\frac {a^2 c \sqrt {c x^2}}{b^3 x (a+b x)}-\frac {2 a c \sqrt {c x^2} \log (a+b x)}{b^3 x} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.81 \[ \int \frac {\left (c x^2\right )^{3/2}}{x (a+b x)^2} \, dx=\frac {c^2 x \left (-a^2+a b x+b^2 x^2-2 a (a+b x) \log (a+b x)\right )}{b^3 \sqrt {c x^2} (a+b x)} \]
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Time = 0.14 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.91
method | result | size |
default | \(-\frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (2 \ln \left (b x +a \right ) x a b -b^{2} x^{2}+2 a^{2} \ln \left (b x +a \right )-a b x +a^{2}\right )}{x^{3} b^{3} \left (b x +a \right )}\) | \(62\) |
risch | \(\frac {c \sqrt {c \,x^{2}}}{b^{2}}-\frac {a^{2} c \sqrt {c \,x^{2}}}{b^{3} x \left (b x +a \right )}-\frac {2 a c \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{b^{3} x}\) | \(63\) |
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Time = 0.22 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.93 \[ \int \frac {\left (c x^2\right )^{3/2}}{x (a+b x)^2} \, dx=\frac {{\left (b^{2} c x^{2} + a b c x - a^{2} c - 2 \, {\left (a b c x + a^{2} c\right )} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{b^{4} x^{2} + a b^{3} x} \]
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\[ \int \frac {\left (c x^2\right )^{3/2}}{x (a+b x)^2} \, dx=\int \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{x \left (a + b x\right )^{2}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.44 \[ \int \frac {\left (c x^2\right )^{3/2}}{x (a+b x)^2} \, dx=\frac {\sqrt {c x^{2}} a c}{b^{3} x + a b^{2}} - \frac {2 \, \left (-1\right )^{\frac {2 \, c x}{b}} a c^{\frac {3}{2}} \log \left (\frac {2 \, c x}{b}\right )}{b^{3}} - \frac {2 \, \left (-1\right )^{\frac {2 \, a c x}{b}} a c^{\frac {3}{2}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{3}} + \frac {\sqrt {c x^{2}} c}{b^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.85 \[ \int \frac {\left (c x^2\right )^{3/2}}{x (a+b x)^2} \, dx=c^{\frac {3}{2}} {\left (\frac {x \mathrm {sgn}\left (x\right )}{b^{2}} - \frac {2 \, a \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{3}} + \frac {{\left (2 \, a \log \left ({\left | a \right |}\right ) + a\right )} \mathrm {sgn}\left (x\right )}{b^{3}} - \frac {a^{2} \mathrm {sgn}\left (x\right )}{{\left (b x + a\right )} b^{3}}\right )} \]
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Timed out. \[ \int \frac {\left (c x^2\right )^{3/2}}{x (a+b x)^2} \, dx=\int \frac {{\left (c\,x^2\right )}^{3/2}}{x\,{\left (a+b\,x\right )}^2} \,d x \]
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